Question

What is the price of a European put option on a non-dividend-paying stock when the stock price is $100, the strike price is $90, the risk- free interest rate is $5% per annum, the volatility is 35% per annum (continuously compounded), and the time to maturity is 6 months? Use the Black-Scholes-Merton option pricing formula.

One second later, the stock is traded at 101. How would you estimate the new price for the option without the Black-Scholes-Merton option pricing formula?

Answer 1

Using Black Scholes model,we have:

Formulas:

d1 = {ln(100/90) + (5% + 35%^2/2)*0.5}/(35%*0.5^0.5) = 0.6505

d2 = 0.6505 - (35%*0.5^0.5) = 0.4030

N(-d1) (reading from the normal Z-table or using NORMDIST function in excel) = 0.2577

Simialry, N(-d2) = N(-0.4030) = 0.3435

Put price = (90*e^(-5%*0.5)*0.3435) - (100*0.2577) = 4.3806

Output:

Put option price = 4.3806

Put option delta = N(d1) -1 = N(0.6505) -1 = -0.2577

Change in option price = delta*change in stock price

= -0.2577*1 = -0.2577

New put option price = 4.3806 -0.2577 = 4.1229

100.00 90.00 Current stock price (S) Strike price (K) Time until expiration(in years) (t) volatility (s) risk-free rate (1) 0.500 35.0% 5.00%

d1 = {In(S/K) + (r +5^2/2)t}(s(t^0.5)) d2 = d1 - (s(t"0.5)) N(-d1) - Normal distribution of -d1 N(-d2) - Normal distribution of -d2 P = K*(e^(-it))*N(-02) - S*N(-d1)

d1 d2 N(-d1) N(-d2) Put premium (P) 0.6505 0.4030 0.2577 0.3435 4.3806